Intersection Homology & Perverse Sheaves
Intersection Homology & Perverse Sheaves

Author: Laurenţiu G. Maxim

Publisher: Springer Nature

Published: 2019-11-30

Total Pages: 278

ISBN-13: 3030276449

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This textbook provides a gentle introduction to intersection homology and perverse sheaves, where concrete examples and geometric applications motivate concepts throughout. By giving a taste of the main ideas in the field, the author welcomes new readers to this exciting area at the crossroads of topology, algebraic geometry, analysis, and differential equations. Those looking to delve further into the abstract theory will find ample references to facilitate navigation of both classic and recent literature. Beginning with an introduction to intersection homology from a geometric and topological viewpoint, the text goes on to develop the sheaf-theoretical perspective. Then algebraic geometry comes to the fore: a brief discussion of constructibility opens onto an in-depth exploration of perverse sheaves. Highlights from the following chapters include a detailed account of the proof of the Beilinson–Bernstein–Deligne–Gabber (BBDG) decomposition theorem, applications of perverse sheaves to hypersurface singularities, and a discussion of Hodge-theoretic aspects of intersection homology via Saito’s deep theory of mixed Hodge modules. An epilogue offers a succinct summary of the literature surrounding some recent applications. Intersection Homology & Perverse Sheaves is suitable for graduate students with a basic background in topology and algebraic geometry. By building context and familiarity with examples, the text offers an ideal starting point for those entering the field. This classroom-tested approach opens the door to further study and to current research.

Singular Intersection Homology
Singular Intersection Homology

Author: Greg Friedman

Publisher: Cambridge University Press

Published: 2020-09-24

Total Pages: 823

ISBN-13: 1107150744

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The first expository book-length introduction to intersection homology from the viewpoint of singular and piecewise linear chains.

Singular Intersection Homology
Singular Intersection Homology

Author: Greg Friedman

Publisher: Cambridge University Press

Published: 2020-09-24

Total Pages: 824

ISBN-13: 1108895360

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Intersection homology is a version of homology theory that extends Poincaré duality and its applications to stratified spaces, such as singular varieties. This is the first comprehensive expository book-length introduction to intersection homology from the viewpoint of singular and piecewise-linear chains. Recent breakthroughs have made this approach viable by providing intersection homology and cohomology versions of all the standard tools in the homology tool box, making the subject readily accessible to graduate students and researchers in topology as well as researchers from other fields. This text includes both new research material and new proofs of previously-known results in intersection homology, as well as treatments of many classical topics in algebraic and manifold topology. Written in a detailed but expository style, this book is suitable as an introduction to intersection homology or as a thorough reference.

Stratified Morse Theory
Stratified Morse Theory

Author: Mark Goresky

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 279

ISBN-13: 3642717144

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Due to the lack of proper bibliographical sources stratification theory seems to be a "mysterious" subject in contemporary mathematics. This book contains a complete and elementary survey - including an extended bibliography - on stratification theory, including its historical development. Some further important topics in the book are: Morse theory, singularities, transversality theory, complex analytic varieties, Lefschetz theorems, connectivity theorems, intersection homology, complements of affine subspaces and combinatorics. The book is designed for all interested students or professionals in this area.

Topological Invariants of Stratified Spaces
Topological Invariants of Stratified Spaces

Author: Markus Banagl

Publisher: Springer Science & Business Media

Published: 2007-02-16

Total Pages: 266

ISBN-13: 3540385878

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The central theme of this book is the restoration of Poincaré duality on stratified singular spaces by using Verdier-self-dual sheaves such as the prototypical intersection chain sheaf on a complex variety. Highlights include complete and detailed proofs of decomposition theorems for self-dual sheaves, explanation of methods for computing twisted characteristic classes and an introduction to the author's theory of non-Witt spaces and Lagrangian structures.

An Introduction to Intersection Homology Theory, Second Edition
An Introduction to Intersection Homology Theory, Second Edition

Author: Frances Kirwan

Publisher: CRC Press

Published: 2006-06-07

Total Pages: 250

ISBN-13: 9781584881841

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Now more that a quarter of a century old, intersection homology theory has proven to be a powerful tool in the study of the topology of singular spaces, with deep links to many other areas of mathematics, including combinatorics, differential equations, group representations, and number theory. Like its predecessor, An Introduction to Intersection Homology Theory, Second Edition introduces the power and beauty of intersection homology, explaining the main ideas and omitting, or merely sketching, the difficult proofs. It treats both the basics of the subject and a wide range of applications, providing lucid overviews of highly technical areas that make the subject accessible and prepare readers for more advanced work in the area. This second edition contains entirely new chapters introducing the theory of Witt spaces, perverse sheaves, and the combinatorial intersection cohomology of fans. Intersection homology is a large and growing subject that touches on many aspects of topology, geometry, and algebra. With its clear explanations of the main ideas, this book builds the confidence needed to tackle more specialist, technical texts and provides a framework within which to place them.

Intersection Theory
Intersection Theory

Author: W. Fulton

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 483

ISBN-13: 3662024217

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From the ancient origins of algebraic geometry in the solution of polynomial equations, through the triumphs of algebraic geometry during the last two cen turies, intersection theory has played a central role. Since its role in founda tional crises has been no less prominent, the lack of a complete modern treatise on intersection theory has been something of an embarrassment. The aim of this book is to develop the foundations of intersection theory, and to indicate the range of classical and modern applications. Although a comprehensive his tory of this vast subject is not attempted, we have tried to point out some of the striking early appearances of the ideas of intersection theory. Recent improvements in our understanding not only yield a stronger and more useful theory than previously available, but also make it possible to devel op the subject from the beginning with fewer prerequisites from algebra and algebraic geometry. It is hoped that the basic text can be read by one equipped with a first course in algebraic geometry, with occasional use of the two appen dices. Some of the examples, and a few of the later sections, require more spe cialized knowledge. The text is designed so that one who understands the con structions and grants the main theorems of the first six chapters can read other chapters separately. Frequent parenthetical references to previous sections are included for such readers. The summaries which begin each chapter should fa cilitate use as a reference.

Topology of Singular Spaces and Constructible Sheaves
Topology of Singular Spaces and Constructible Sheaves

Author: Jörg Schürmann

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 461

ISBN-13: 3034880618

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This volume is based on the lecture notes of six courses delivered at a Cimpa Summer School in Temuco, Chile, in January 2001. Leading experts contribute with introductory articles covering a broad area in probability and its applications, such as mathematical physics and mathematics of finance. Written at graduate level, the lectures touch the latest advances on each subject, ranging from classical probability theory to modern developments. Thus the book will appeal to students, teachers and researchers working in probability theory or related fields.

Handbook of Geometry and Topology of Singularities II
Handbook of Geometry and Topology of Singularities II

Author: José Luis Cisneros-Molina

Publisher: Springer Nature

Published: 2021-11-01

Total Pages: 581

ISBN-13: 3030780244

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This is the second volume of the Handbook of the Geometry and Topology of Singularities, a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research. This volume consists of ten chapters which provide an in-depth and reader-friendly survey of some of the foundational aspects of singularity theory and related topics. Singularities are ubiquitous in mathematics and science in general. Singularity theory interacts energetically with the rest of mathematics, acting as a crucible where different types of mathematical problems interact, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject, and in other subjects. Authored by world experts, the various contributions deal with both classical material and modern developments, covering a wide range of topics which are linked to each other in fundamental ways. The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.